Anglo-Saxon alliterative verse is basically a more primitive version of the forms used in Nordic poetry and described in the former chapter; trusting you haven't read it, scared as you were by the ø's, þ's and å's, the Present Author recommends you go back and consult it: it is mostly in English.

A short historyAlliterative verse was used somewhat properly, and very abundantly, in Old English; it was still used in Middle English,
but the 'liberties' granted to the writer there were such that alliteration was little more than a scattered, occasional device in the text, and the rhymed, syllabic forms imported from Southern Europe by Chaucer and others soon overcame the last remnants of these forms.
As 'national identity' became a factor in the Romantic age and British poets began to cherish the idea they hadn't spent the middle ages living in caves and eating acorns, some of them, and mostly Coleridge, attempted what they fancied to be alliterative verses, producing none, but creating some interesting pieces nonetheless. Lastly, in the recently dead 20th century, poets such as Ted Hughes and Pound have resumed writing pieces more or less as in the old times, and that's just as good as poetry gets today.

Alliteration (definition)
Whatever your favourite dictionary might say about the matter, alliteration in ancient Nordic and English poetry
is defined as the identity of the first sounds preceding the vowels in the syllables carrying the primary stresses of two words; examples are 'but-tocks' and 'bran-ding' or 'sod-o-mise' and 'con-cern'. Also, in most works of this type, this definition is valid if there is no such sound, as in the 'arse' and 'hon-est' couple.
I'd better remind the usual pests that a word can have only one primary stress, so that the definition lends itself to no ambiguity.

Versification based on beatsThis technique is a commonly mentioned myth in modern poetry, but it was actually used in alliterative verse,
and (with some further restrictions, the topic shan't be touched here) in classic Latin poetry; its origin is extremely old, as it was already used in the Middle East in the 20th century b.C., and perhaps even before. It is the only alternative to syllabic count practised before free verse, so it deserves some respect; it consists in writing verses which have a fixed number of clearly perceivable main stresses, and the true problem in writing them is making their nature and position unmistakeable.
A major drawback of this otherwise brilliant metric idea is that a sequence of lines written in this way, and with the
same beats count, tends to sound as dull as a Swiss Sunday; Blake resorts often to this technique to try to give some unity to his (slightly) unruly verse length. It is, in this Author's opinion, just a coincidence. There is purpose, however, in 'A crocodile' by Thomas Lovell Beddoes (next time you wonder 'what's in a name?' think of this one):

HARD by the lilied Nile I saw
A duskish river-dragon stretched along,
The brown habergeon of his limbs enamelled
With sanguine almandines and rainy pearl:
And on his back there lay a young one sleeping,
No bigger than a mouse; with eyes like beads,
And a small fragment of its speckled egg
Remaining on its harmless, pulpy snout;
A thing to laugh at, as it gaped to catch
The baulking merry flies. In the iron jaws
Of the great devil-beast, like a pale soul
Fluttering in rocky hell, lightsomely flew
A snowy trochilus, with roseate beak
Tearing the hairy leeches from his throat

(this example has four stresses per line, for the most distracted among you).

The alliterative verseThe origin of alliterative verse would be obvious to anyone that:

has read chapter 6 of this manual

has noticed how Swedes, Danes, Icelanders and Norses read Old English with much more ease than any native Englishman

realises that linebreaks had no importance in ancient poetry

As these conditions are not verified simultaneously in any being on Earth except for the Present Author, he shall enlighten you on the point: take for example fornyrðislag: if two verses of this form are crammed in the same line, the result is a single verse with two beats before a caesura and two beats after it; moreover, alliteration (in the sense defined above) will connect the first three stress-bearing words: and that is exactly how alliterative verse is built in Old English. An example is the first of the Exeter riddles (which the Present Author cannot even read, imagine solve, but which illustrates the point nicely):

has lines that can be split in two without loss of 'poetic identity' or whatever other virtue Farmer Joe attributes to linebreaks when preparing his diary for the editor.

Asking to adhere to such a strict form century after century of a population which, for the most part, did
actually feed on acorns and live in caves, though, seemed to be impossible, and the readers with the keenest eyes will have probably noticed some of the lines of the Exeter riddle do not comply fully to the metre as it is described. The first thing to be removed from the complete aa|ax (where letters now denote alliterations rather than rhyming as in chapters 2-5) was the fact that all the first three beats should be preceded by alliterating sounds: in 'Beowulf' and other longish sagas, the rule is
slackened and the first beat of the second hemistich has to alliteratively match only one of those of the first, practically leading to three possibilities:

aa|ax

ay|ax

ya|ax

Later involutionAfter the transition from Old English
to the actually readable Middle English, the metres we are concerned with here begin to be used for longer works, and in an even more relaxed way: the number of beats can indifferently be four or five, (in the latter case the caesura, still mandatory, alls after the third beat) and the patterns of alliteration within the verse offer many more choices than before, including ones that involve
separate alliterative groups within the same line; a list, probably incomplete, could be this:

Actually, the situation was not so tragic as it might seem, and most of the structures above are quite rare in comparison
to the traditional ones, but the seed of anarchy, in a sense, was cast. In addition to that, an increasing interest turned, in the age of Middle English, towards the end of the verse: Langland's 'Piers Plowman' has only feminine lines, while the anonymous 'Pearl' is heavily rimed: the end of alliterative verse, as king Herod's, came from its toes.